REPORT ON LOCALLY FINITE TRIANGULATED CATEGORIES

HENNING KRAUSE

Abstract. The basic properties of locally finite triangulated categories are discussed. The focus is on Auslander–Reiten theory and the lattice of thick subcategories.

Contents

1. Introduction  1
2. Locally noetherian triangulated categories  2
3. Auslander–Reiten theory for triangulated categories  6
4. The lattice of thick subcategories  9
5. Simply connected triangulated categories  12
6. Thick subcategories and non-crossing partitions  13
Appendix A. Auslander–Reiten theory  18
References  24

1. Introduction

This is a report on a particular class of triangulated categories. A triangulated category T is said to be locally finite if every cohomological functor from T or its opposite category T^{op} into the category of abelian groups is a direct sum of representable functors.

We present a number of basic results for such triangulated categories. Some of these results seem to be new, but we include also results which are variations or generalisations of known results. Thus our aim is to provide the foundations for studying the locally finite triangulated categories.

A basic tool for understanding a triangulated category T is the collection of representable functors Hom_{T}(−, X): T^{op} → Ab where X runs through the objects of T. We show that T is locally finite if and only if each representable functor is of finite length. This property justifies the term ‘locally finite’ which is due to Xiao and Zhu [51] in the triangulated context and goes back to Gabriel [21].

An important thread in the study of locally finite triangulated categories is the use of Auslander–Reiten theory. The principal idea is to analyse for each object X in T the radical filtration

… ⊆ Rad_{T}^{2}(−, X) ⊆ Rad_{T}^{1}(−, X) ⊆ Rad_{T}^{0}(−, X) = Hom_{T}(−, X)

which is finite when T is locally finite. Some of this information is encoded in the Auslander–Reiten quiver of T which can be described fairly explicitly.

Another intriguing invariant of a triangulated category is the lattice of thick subcategories. Assuming locally finiteness, we show that the inclusion of each thick subcategory admits a left and a right adjoint. In fact, the lattice has interesting symmetries and is even finite if the category is finitely generated.

Version from January 17, 2011.

The results presented here are most complete when the category is simply connected, that is, the Auslander–Reiten quiver is connected and contains no oriented cycle. For instance, the lattice of thick subcategories is in this case isomorphic to the lattice of non-crossing partitions associated to some diagram of Dynkin type.

Using covering theory, the study of a general locally finite triangulated category can often be reduced to the simply connected case. For this direction we refer to recent work of Amiot [1] and Köhler [36].

Acknowledgements. It is a pleasure to thank numerous colleagues for interesting discussions and helpful comments on the subject of this work. Let me mention explicitly Apostolos Beligiannis, Otto Kerner, Claudia Köhler, Shiping Liu, Claus Ringel, Jan Šťovíček, Hugh Thomas, and Dieter Vossieck.

2. Locally noetherian triangulated categories

In this section, locally noetherian and locally finite triangulated categories are introduced. We provide various characterisations and a host of examples. Then we establish the existence of adjoints for inclusions of thick subcategories.

Throughout this work let T denote a triangulated category with suspension Σ.

The abelianisation of a triangulated category. Following Freyd [20, §3] and Verdier [48, II.3], we consider the abelianisation A(T) of a triangulated category T which is by definition the abelian category consisting of all additive functors F: T^{op} → Ab into the category of abelian groups that fit into an exact sequence

Hom_{T}(−, X) ⟶ Hom_{T}(−, Y) ⟶ F ⟶ 0.

The fully faithful Yoneda functor H: T → A(T) taking an object X to the representable functor Hom_{T}(−, X) is the universal cohomological functor starting in T, that is, each cohomological functor T → A to an abelian category A factors essentially uniquely through H. Observe that the functor taking Hom_{T}(−, X) to Hom_{T}(X, −) induces an equivalence

(2.1)  A(T)^{op} ⟶^{∼} A(T^{op}).

This is an immediate consequence of the universal property of the Yoneda functor.

Locally noetherian triangulated categories. Given an essentially small triangulated category T, we use its abelianisation to formulate a useful finiteness condition; see also [9]. We say that T is locally noetherian^{1} if the equivalent conditions of the following theorem are satisfied.

Theorem 2.1. For an essentially small triangulated category T the following conditions are equivalent.

(1) Every cohomological functor T^{op} → Ab into the category of abelian groups is a direct sum of representable functors.

(2) Every object in T is a finite coproduct of indecomposable objects with local endomorphism rings, and for every sequence X_{1} ⟶^{ϕ_{1}} X_{2} ⟶^{ϕ_{2}} X_{3} ⟶^{ϕ_{3}} … of non-isomorphisms between indecomposable objects there exists some number n such that ϕ_{n} … ϕ_{2}ϕ_{1} = 0.

1The terminology refers to the equivalent fact that the abelian category of additive functors T^{op} → Ab is locally noetherian in the sense of [21, II.4].

(3) Idempotents in T split and every object of the abelianization A(T) is noetherian, that is, it satisfies the ascending chain condition on subobjects.

Proof. We view the additive category T as a ring with several objects and think of additive functors T^{op} → Ab as T-modules. Note that a T-module is flat if and only if it is a cohomological functor; see [37, Lemma 2.7]. Bass has characterised the rings for which every flat module is projective. This can be generalised to modules over rings with several objects, see [30, Theorem B.12], and yields the equivalence of conditions (1) and (2).

Recall that a module M is fp-injective if Ext^{1}(−, M) vanishes on all finitely presented modules. Note that T as a ring with several objects is noetherian (that is, each representable functor Hom_{T}(−, X) satisfies the ascending chain condition on subfunctors) if and only if every fp-injective T-module is injective; see [30, Theorem B.17]. The fp-injective T-modules are precisely the cohomological functors T^{op} → Ab, by [37, Lemma 2.7].

Suppose that (1) holds and fix an fp-injective T-module M. Choose an injective envelope ϕ: M → Q. The snake lemma shows that the cokernel Coker ϕ is cohomological. Thus Coker ϕ is a direct sum of representable functors and therefore projective; in particular ϕ splits. It follows that M is injective, and therefore T is noetherian. Thus (3) holds.

Now suppose that (3) holds. Thus T is noetherian. Fix a flat T-module M and choose an epimorphism ϕ: P → M such that P is projective. The snake lemma shows that the kernel Ker ϕ is cohomological. Thus Ker ϕ is fp-injective and therefore injective; in particular ϕ splits. It follows that M is projective, and therefore a direct sum of finitely generated projective modules; see [30, Corollary B.13]. The finitely generated projective modules are precisely the representable functors since T has split idempotents. Thus (1) holds. □

Example 2.2. Fix a field k and denote by A the category of k-linear representations of the quiver

Γ: 1 ⟶ 2 ⟶ 3 ⟶ 4 ⟶ ⋯

that are finite dimensional and have finite support. Then the bounded derived category D^{b}(A) is locally noetherian but its opposite category is not.

Indeed, each object in D^{b}(A) decomposes into a finite coproduct of indecomposable objects with local endomorphism rings. The indecomposable objects are isomorphic to complexes concentrated in a single degree (thus of the form X[i] with X ∈ A and i ∈ ℤ) since Ext_{A}^{p}(−, −) = 0 for p > 1.

Given a sequence X_{1}, …, X_{r} of objects in A and a sequence of morphism X_{1}[i_{1}] → X_{2}[i_{2}] → ⋯ → X_{r}[i_{r}] in D^{b}(A) such that their composite is non-zero, we have i_{1} ≤ … ≤ i_{r} ≤ i_{1} + 1. Thus it remains to observe that infinite chains of nonisomorphisms between indecomposable objects in A exist only in one direction. More precisely, for each n ≥ 1 let A_{n} denote the full subcategory of representations with support contained in {1, …, n}. Then any non-zero morphism X → Y betweeen indecomposable representations has the property that X ∈ A_{n} implies Y ∈ A_{n}. This is an immediate consequence of the fact that each indecomposable representation of Γ is, up to isomorphism, of the following form:

0 → ⋯ → 0 → k ⟶^{1} ⋯ ⟶^{1} k → 0 → ⋯

On the other hand, there is an obvious chain of proper epimorphisms ⋯ → X_{3} → X_{2} → X_{1} in A, where X_{n} denotes the unique indecomposable representation with support {1, …, n}.

Locally finite triangulated categories. An essentially small triangulated category T is said to be locally finite if T and T^{op} are locally noetherian. Locally finite means that each representable functor Hom_{T}(−, X) is of finite length as a T-module. This follows from the equivalence A(T)^{op} →^{∼} A(T^{op}). In particular, T is locally finite if and only if the category of T-modules is locally finite in the sense of [21, II.4].

Suppose that T is locally finite and fix an object Y. Then there are only finitely many isomorphism classes of indecomposable objects X satisfying Hom_{T}(X, Y) ≠ 0, because Hom_{T}(X, Y) ≠ 0 implies that Hom_{T}(−, X) is the projective cover of a composition factor of Hom_{T}(−, Y). This observation gives rise to the following characterisation which can be deduced from [3, Theorem 2.12].

Proposition 2.3 (Auslander). An essentially small triangulated category T with split idempotents is locally finite if and only if for each object Y the following holds:

(1) There are only finitely many isomorphism classes of indecomposable objects X satisfying Hom_{T}(X, Y) ≠ 0.

(2) For each indecomposable object X, the End_{T}(X)-module Hom_{T}(X, Y) is of finite length. □

Examples. We list some examples of triangulated categories that are locally finite. Throughout we fix a field k.

(1) Let T be an essentially small k-linear triangulated category. Suppose that idempotents split and that morphism spaces are finite dimensional. Then T is locally finite if and only if for each object Y there are only finitely many isomorphism classes of indecomposable objects X satisfying Hom_{T}(X, Y) ≠ 0. This follows from Proposition 2.3 and serves as a definition in [51].

(2) Let A be a finite dimensional k-algebra and suppose that k is algebraically closed. Then the bounded derived category D^{b}(mod A) of the category of finite dimensional A-modules is locally finite if and only if it is triangle equivalent to D^{b}(mod kΓ) for some path algebra kΓ of a finite quiver Γ such that its underlying diagram is a disjoint union of diagrams of Dynkin type; see [25, §5] and [10, Theorem 12.20].

(3) Let A be an essentially small hereditary abelian category. Then the derived category D^{b}(A) is locally finite if and only if A satisfies the conditions in Proposition 2.3. This follows from the fact that each indecomposable object is isomorphic to a complex that is concentrated in a single degree. If A is the category of finitely generated modules over an artinian ring, then this condition means that the ring is of finite representation type.

(4) Let A be a noetherian ring and suppose that A is Gorenstein, that is, A has finite injective dimension as an A-module. Denote by MCM(A) the category of finitely generated A-modules X that are maximal Cohen–Macaulay, which means that Ext_{A}^{i}(X, A) = 0 for all i > 0. This is an exact Frobenius category, and the stable category \underline{MCM}(A) modulo all morphisms that factor through a projective object is a triangulated category [16].

If A is a finite dimensional and self-injective k-algebra, then all A-modules are maximal Cohen–Macaulay, and \underline{MCM}(A) is locally finite if and only if A is of finite representation type, that is, there are only finitely many isomorphism classes of indecomposable A-modules. These algebras have been classified [44, 49].

If A is a commutative complete local ring, then A is by definition of finite Cohen– Macaulay type if there there are only finitely many isomorphism classes of indecomposable maximal Cohen–Macaulay modules over A. In that case \underline{MCM}(A) is locally finite. There is a whole theory describing such rings and a parallel theory for graded Gorenstein algebras; see [50].

(5) Let Γ be a quiver of Dynkin type. Then the orbit category D^{b}(mod kΓ)/G of the derived category with respect to an appropriate group G of automorphisms is a locally finite triangulated category [33, 1]. Examples are the cluster categories of finite type [15].

(6) The category of finitely generated projective modules over the ring ℤ/4ℤ carries a triangulated structure that admits no model [42]; it is a locally finite triangulated category.

Orthogonal subcategories. Let T be a triangulated category and S a triangulated subcategory. Then we define two full subcategories

S^{⊥} = {Y ∈ T | Hom_{T}(X, Y) = 0 for all X ∈ S}
⊥S = {X ∈ T | Hom_{T}(X, Y) = 0 for all Y ∈ S}

and call them orthogonal subcategories with respect to S. Note that S^{⊥} and ^{⊥}S are thick subcategories of T.

The following lemma collects some basic facts about orthogonal subcategories which are well-known. For a proof, see [39, Proposition 4.9.1].

Lemma 2.4. Let T be a triangulated category and S a thick subcategory. Then the following are equivalent.

(1) The inclusion functor S → T admits a right adjoint.

(2) The composite S^{⊥} ⟶^{inc} T ⟶^{can} T/S is an equivalence.

(3) The inclusion functor S^{⊥} → T admits a left adjoint and ^{⊥}(S^{⊥}) = S. □

There is an interesting consequence. If S is a thick subcategory of T such that the inclusion admits a left and a right adjoint, then one has equivalences

⊥S ⟶^{∼} T/S ⟵^{∼} S^{⊥}.

Existence of adjoints. Triangulated categories that are locally noetherian have the following remarkable property.

Theorem 2.5. Let T be an essentially small triangulated category and suppose that T is locally noetherian. Then for each thick subcategory of T the inclusion functor admits a right adjoint.

Proof. Fix a thick subcategory U and an object X in T. We need to construct a morphism U → X with U in U inducing a bijection Hom_{T}(U′, U) → Hom_{T}(U′, X) for all U′ in U. Take the comma category U/X consisting of all morphisms U →^{ϕ} X

with U in U. A morphism from U →^{ϕ} X to U′ →^{ϕ′} X is a morphism μ: U → U′ such that ϕ′μ = ϕ. This category is closely related to Verdier’s construction of the localisation functor T → T/U; see [48, II.2]. The arguments given there show that U/X is filtered. Therefore the functor

colim_{U→X} Hom_{T}(−, U)

is cohomological. Moreover, one obtains an exact sequence of cohomological functors

⋯ ⟶ Hom_{T/U}(−, Σ^{−1}X) ⟶ colim_{U→X} Hom_{T}(−, U) ⟶

⟶ Hom_{T}(−, X) ⟶ Hom_{T/U}(−, X) ⟶ ⋯ since one has by definition

Hom_{T/U}(−, X) = colim_{X→V} Hom_{T}(−, V)

where X → V runs through all morphisms with cone in U.

Now we use that T is locally noetherian and write

colim_{U→X} Hom_{T}(−, U) = ⨁_{i∈I} Hom_{T}(−, U_{i})

as a direct sum of representable functors. Similarly, we get

Hom_{T/U}(−, X) = ⨁_{j∈J} Hom_{T}(−, V_{j}).

We may assume that U_{i} and V_{j} are non-zero for all i, j. Observe that U_{i} ∈ U and V_{j} ∈ U^{⊥} for all i, j. The morphism

Hom_{T}(−, X) ⟶ ⨁_{j∈J} Hom_{T}(−, V_{j})

factors through a finite sum ⨁_{j∈J_{0}} Hom_{T}(−, V_{j}). In fact, the exactness of the above sequence implies that V_{j} belongs to U for each j ∈ J ∖ J_{0}. Thus J = J_{0} and therefore Hom_{T/U}(−, X) belongs to A(T). It follows that I is also finite, since colim_{U→X} Hom_{T}(−, U) is an extension of two objects in A(T). This yields a morphism U = ∐_{i} U_{i} → X inducing a bijection Hom_{T}(U′, U) → Hom_{T}(U′, X) for all U′ in U. □

Corollary 2.6. Let T be an essentially small triangulated category and suppose that T is locally noetherian. If U is a thick subcategory of T, then ^{⊥}(U^{⊥}) = U.

Proof. Clearly, ^{⊥}(U^{⊥}) contains U. Now pick an object X in ^{⊥}(U^{⊥}). Let U → X be the universal morphism from an object in U to X, and complete this to an exact triangle U → X → V → ΣU. Then V belongs to U^{⊥} and therefore Hom_{T}(X, V) = 0. It follows that X belongs to U. □

The following example shows that the identity ^{⊥}(U^{⊥}) = U does not hold in general.

Example 2.7. Consider the bounded derived category D^{b}(mod ℤ) of finitely generated modules over the ring ℤ of integers. The complexes with torsion cohomology form a thick subcategory U = D_{tor}^{b}(mod ℤ) such that ^{⊥}U = 0.

Corollary 2.8. Let T be an essentially small triangulated category and suppose that T is locally noetherian. If U a thick subcategory of T, then the categories U and T/U are locally noetherian.

Proof. The inclusion U → T induces a fully faithful and exact functor A(U) → A(T). Thus every object of A(U) is noetherian. On the other hand, there is an equivalence U^{⊥} ⟶^{∼} T/U by Lemma 2.4. Thus T/U is locally noetherian, since U^{⊥} is locally noetherian. □

3. Auslander–Reiten theory for triangulated categories

We describe briefly the Auslander–Reiten theory for an essentially small triangulated category T that is locally noetherian. For general concepts from Auslander– Reiten theory we refer to Appendix A.

Auslander–Reiten triangles. Recall from [24] that an exact triangle

X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

is an Auslander–Reiten triangle starting at X and ending at Z if (1) α is a left almost split morphism,

(2) β is a right almost split morphism,

(3) γ ≠ 0. Observe that these properties hold if and only if the morphism β is minimal right almost split, by [38, Lemma 2.6].

Proposition 3.1. Given an essentially small triangulated category that is locally noetherian, there exists for each indecomposable object an Auslander–Reiten triangle ending at it.

Proof. Fix an indecomposable object Z in T. Then the simple T-module S_{Z} = Hom_{T}(−, Z)/ Rad_{T}(−, Z) is finitely presented since T is locally noetherian. Thus we can choose in A(T) a minimal projective presentation

Hom_{T}(−, Y) ⟶ Hom_{T}(−, Z) ⟶ S_{Z} ⟶ 0;

see Lemma A.1. It follows from Lemma A.5 that the induced morphism Y → Z is minimal right almost split. Completing this morphism to an exact triangle yields an Auslander–Reiten triangle ending at Z. □

The definition of an Auslander–Reiten triangle is symmetric. Thus there are Auslander–Reiten triangles in T starting at each indecomposable object if T^{op} is locally noetherian. This gives the existence of Auslander–Reiten triangles for locally finite triangulated categories. For compactly generated triangulated categories, this result is due to Beligiannis [11, Theorem 10.2].

Corollary 3.2. Given an essentially small triangulated category that is locally finite, there exist Auslander–Reiten triangles starting and ending at each indecomposable object. □

Remark 3.3. Let X → Y → Z → ΣX be an Auslander–Reiten triangle in an essentially small triangulated category that is locally noetherian. The relation between the end terms can be explained as follows. Let A = End_{T}(Z) and denote by E = E(A/ rad A) an injective envelope. Then

(3.1)  Hom_{A}(Hom_{T}(Z, −), E) ≅ Hom_{T}(−, ΣX);

see [38, Theorem 2.2]. In particular, End_{T}(X) ≅ End_{A}(E).

The Auslander–Reiten quiver. For a locally finite triangulated category the structure of the Auslander–Reiten quiver has been determined in work of Xiao and Zhu [51] and Amiot [1]. In fact, these authors consider triangulated categories that are linear over a field with finite dimensional morphism spaces. Then they apply the structural results on valued translation quivers due to Riedtmann [43] and Happel, Preiser, and Ringel [26]. The same arguments work in a slightly more general settig, thanks to the following result; see also [41, Proposition 2.1].

For the definition of a valued translation quiver, see [41, §2]. The original definition [27, §2] excludes loops, but they are possible in our setting.

Proposition 3.4. Let k be a commutative ring and T an essentially small k-linear triangulated category such that all morphism spaces are of finite length over k. Suppose that T is locally finite. Then the Auslander–Reiten quiver of T is a valued translation quiver. Assigning to a vertex X the length ℓ(X) of Hom_{T}(−, X) in the abelianisation A(T) yields a subadditive function on the set of vertices such that for each vertex Z

2ℓ(Z) = ℓ(Z) + ℓ(τZ) = 2 + ∑_{Y →Z} δ_{Y,Z}ℓ(Y).

Proof. The existence of Auslander–Reiten triangles has already been established, and this gives the translation τ. The identities for the valuation are precisely the statements of Lemmas A.9 and A.10. For the second part one uses the fact that each Auslander–Reiten triangle τZ → Ȳ → Z → Σ(τZ) induces an exact sequence

0 → S_{Σ^{−1}Z} → Hom_{T}(−, τZ) → Hom_{T}(−, Ȳ) → Hom_{T}(−, Z) → S_{Z} → 0

in A(T) by Lemma A.5. Then one applies Lemma A.6 which gives a decomposition

Ȳ = ∐_{Y →Z} Y^{δ_{Y,Z}}

where Y → Z runs through all arrows ending at Z. It remains to observe that ℓ(Z) = ℓ(τZ). This follows from the isomorphism (3.1) and an appropriate description of ℓ, see [3, Proposition 2.11]. Note that Hom_{T}(−, X) and Hom_{T}(X, −) have the same length, thanks to the duality (2.1). □

Theorem 3.5 (Xiao–Zhu). Let k be a commutative ring and T an essentially small k-linear triangulated category such that all morphism spaces are of finite length over k. Suppose that T is locally finite. Then each connected component of the Auslander–Reiten quiver of T is of the form ℤΔ/G for some tree Δ of Dynkin type and some group G of automorphisms of ℤΔ.

Proof. Adapt the proofs of [51, Theorem 2.3.4] or [1, Theorem 4.0.4], using Proposition 3.4. □

Example 3.6. Let k be a field and Γ be a quiver of Dynkin type. Then the Auslander–Reiten quiver of D^{b}(mod kΓ) is of the form ℤΓ; see [24, §4].

A decomposition C = C_{1} ⨿ C_{2} of an additive category C is a pair of full additive subcategories C_{1} and C_{2} such that each object in C is a direct sum of two objects from C_{1} and C_{2}, and Hom_{C}(X_{1}, X_{2}) = 0 = Hom_{C}(X_{2}, X_{1}) for all X_{1} ∈ C_{1} and X_{2} ∈ C_{2}. An additive category C is connected if it admits no proper decomposition C = C_{1} ⨿ C_{2}.

Proposition 3.7. Let T be an essentially small triangulated category that is locally finite. Then any non-zero morphism X → Y between two indecomposable objects in T gives a path X → ⋯ → Y in the Auslander–Reiten quiver of T. In particular, the category T is connected if and only if its Auslander–Reiten quiver is connected.

Proof. We prove the first statement; the second statement is then an immediate consequence. Let ϕ: X → Y be a non-zero morphism. If ϕ is invertible, then the path between X and Y has length zero. Otherwise, ϕ factors through the right almost split morphism ending at Y_{0} = Y, which exists by Proposition 3.1. It follows from Lemma A.6 that there is an arrow Y_{1} → Y_{0} and a non-zero morphism X → Y_{0} that factors through an irreducible morphism Y_{1} → Y_{0}. We continue with the corresponding morphism X → Y_{1}, and the process terminates since T is locally finite. □

It is interesting to note that the Auslander–Reiten quiver of T can be identified with the Ext-quiver [22, 7.1] of the abelian length category A(T), using the bijection from Lemma A.3 between the indecomposable objects of T and the simple objects of A(T). For the bijection between arrows one uses Lemma A.4.

The Nakayama functor. Let k be a field and T an essentially small k-linear triangulated category with finite dimensional morphism spaces. Suppose that T is locally finite. Then there is for each object X in T an object NX representing the k-dual of Hom_{T}(X, −), that is,

Hom_{k}(Hom_{T}(X, −), k) ≅ Hom_{T}(−, NX).

More precisely, we have an isomorphism

Hom_{k}(Hom_{T}(X, −), k) ≅ ⨁_{i∈I} Hom_{T}(−, Y_{i})

for some collection of indecomposable objects Y_{i} since T is locally noetherian. It follows from Proposition 2.3 that I is finite. Thus NX = ∐_{i} Y_{i} is a representing object. This gives a functor N: T → T which is known as Nakayama functor in representation theory [23], or as Serre functor in algebraic geometry [13]. It is easily checked that this functor is an equivalence; a quasi-inverse is given by the Nakayama functor for T^{op} which sends an object X to the object representing Hom_{k}(Hom_{T}(−, X), k). The exactness then follows from [13, Proposition 3.3] or [47, Theorem A.4.4].

Note that for each indecomposable object Z in T, one obtains an Auslander– Reiten triangle X → Y → Z → ΣX by first choosing a non-zero k-linear map End_{T}(Z) → k annihilating the unique maximal ideal of End_{T}(Z) and then completing the corresponding morphism Z → NZ to an exact triangle Σ^{−1}(NZ) → Y → Z → NZ; see [38] for details.

4. The lattice of thick subcategories

Let T be an essentially small triangulated category. We denote by T(T) the set of all thick subcategories of T. This set is partially ordered by inclusion. Observe that for any collection of thick subcategories U_{i} its intersection ⋂_{i} U_{i} is thick. Thus T(T) is a lattice, that is, we can form for each pair of thick subcategories U, V its infimum U ∧ V = U ∩ V and its supremum U ∨ V = ⋂_{S} S, where S runs through all thick subcategories containing U and V. In fact, the lattice T(T) is complete since for any collection of thick subcategories U_{i} its infimum ⋀_{i} U_{i} and its supremum ⋁_{i} U_{i} exist.

If X is an object or collection of objects in T, we write Thick(X) for the thick subcategory generated by X, that is, the smallest thick subcategory containing T. The triangulated category T is finitely generated if there is some object X such that T = Thick(X).

Compactness. Let L be a lattice. An element x of L is compact if x ≤ ⋁_{i∈I} y_{i} implies x ≤ ⋁_{i∈J} y_{i} for some finite subset J ⊆ I. The lattice L is compact if it has a greatest element that is compact.

Lemma 4.1. The lattice T(T) is compact if and only if T is finitely generated. □

Noetherianess. A lattice L is noetherian if there is no infinite strictly increasing chain x_{0} < x_{1} < x_{2} < … in L.

Proposition 4.2. Let T be an essentially small triangulated category that is finitely generated and locally noetherian. Then the lattice T(T) is noetherian.

Proof. Let T = Thick(X) and write H = Hom_{T}(−, X). For each thick subcategory U let X_{U} → X be the universal morphism from an object in U ending at X, which exists by Theorem 2.5. Note that U = Thick(X_{U}) since T = Thick(X). Denote by H_{U} the image of the induced morphism Hom_{T}(−, X_{U}) → H in A(T), and observe that the induced morphism Hom_{T}(−, X_{U}) → H_{U} is a projectice cover.

Let V be another thick subcategory of T. Then H_{U} = H_{V} implies U = V. Indeed, H_{U} = H_{V} implies that their projectice covers are isomorphic. Thus X_{U} ≅ X_{V}, and therefore

U = Thick(X_{U}) = Thick(X_{V}) = V.

Clearly, U ⊆ V implies H_{U} ⊆ H_{V}. It follows that T(T) is noetherian, since the lattice of subobjects of H in A(T) is noetherian. □

Example 4.3. Recall from Example 2.2 that representations of the quiver

1 ⟶ 2 ⟶ 3 ⟶ 4 ⟶ ⋯

give rise to a triangulated category D^{b}(A) that is locally noetherian. The full subcategories A_{n} ⊆ A consisting of all representations with support contained in {1, …, n} yield an infinite strictly increasing chain D^{b}(A_{1}) ⊆ D^{b}(A_{2}) ⊆ … of thick subcategories in D^{b}(A).

Complements. Assigning to a thick subcategory U its orthogonal subcategories U^{⊥} and ^{⊥}U yields two order reversing maps T(T) → T(T). These maps are of interest because we have for two objects X, Y in T

Hom_{T}^{∗}(X, Y) = 0 ⟺ Thick(X)^{⊥} ≥ Thick(Y)

where

Hom_{T}^{∗}(X, Y) = ⨁_{n∈ℤ} Hom_{T}(X, Σ^{n}Y).

Let us collect the basic properties of both maps when T is locally finite.

Proposition 4.4. Let T be an essentially small triangulated category and suppose that T is loccaly finite.

(1) The maps T(T) → T(T) taking U to U^{⊥} and ^{⊥}U are mutually inverse. Thus the lattice T(T) is self-dual.

(2) Let U be a thick subcategory. Then U ∧ U^{⊥} = 0 and U ∨ U^{⊥} = T.

(3) Let V ⊆ U ⊆ T be thick subcategories. Then the quotient U/V is a locally finite triangulated category, and there is a lattice isomorphism

[V, U] = {X ∈ T(T) | V ⊆ X ⊆ U} ⟶^{∼} T(U/V).

The map takes X ⊆ U to X/V.

Proof. (1) We have ^{⊥}(U^{⊥}) = U = (^{⊥}U)^{⊥} by Corollary 2.6.

(2) Clearly, U ∩ U^{⊥} = 0. On the other hand, each object X in T fits into an exact triangle U → X → V → ΣU with U ∈ U and V ∈ U^{⊥}. Thus U ∨ U^{⊥} = T.

(3) The category U/V is locally finite by Corollary 2.8, and the inverse map T(U/V) → [V, U] takes X to its inverse image under the localisation functor U → U/V. □

This proposition says that the lattice is relatively complemented, that is, each interval is complemented. A lattice L is complemented if for each x ∈ L there exists y ∈ L such that x ∨ y = 1 and x ∧ y = 0.

If T is locally finite and admits a Nakayama functor N: T ⟶^{∼} T, then this induces a lattice automorphism of T(T) by taking a thick subcategory U to NU. It follows from the definition of N that NU = (U^{⊥})^{⊥}.

Finiteness. There are further finiteness results for the lattice T(T) if T is locally finite.

Proposition 4.5. Let T be an essentially small triangulated category. If T is finitely generated and locally finite, then T has only finitely many thick subcategories.

Proof. Let T = Thick(X). For each thick subcategory U let X_{U} → X be the universal morphism from an object in U ending at X, which exists by Theorem 2.5. Note that U = Thick(X_{U}) since T = Thick(X). Each indecomposable direct summand X′ of X_{U} satisfies Hom_{T}(X′, X) ≠ 0. There are only finitely many isomorphism classes of such objects by Proposition 2.3. It follows that T(T) is finite. □

Auslander–Reiten theory is used in an essential way for proving the following.

Theorem 4.6. Let T be an essentially small and locally finite triangulated category. If T is connected then T is finitely generated and has therefore only finitely many thick subcategories.

We need the following lemma.

Lemma 4.7 (Amiot). Let T be an essentially small triangulated category that is locally finite. Then any connected component of the Auslander–Reiten quiver of T is – after removing possible loops – of the form ℤΔ/G for some finite tree Δ and some group G of automorphisms of ℤΔ.

Proof. If X → X is an irreducible morphism in T then X = τX, by [1, Proposition 4.1.1]. After removing such arrows X → X, the Auslander–Reiten quiver of T is a stable translation quiver and therefore of the form ℤΔ/G by Riedtmann’s Struktursatz [43]. The tree is finite by [1, Lemma 4.2.2]. □

Proof of Theorem 4.6. The Auslander–Reiten quiver of T is connected by Proposition 3.7 and therefore – modulo loops – of the form ℤΔ/G for some finite tree Δ and some group G of automorphisms by Lemma 4.7. Note that ℤΔ does not depend on the orientation of Δ. Thus we may assume that Δ has no path of length > 1. Then there are two types of vertices. Call a vertex x ∈ Δ_{0} minimal if there is no arrow from Δ ending at x, and maximal if there is no arrow from Δ starting at x.

Let π: ℤΔ → ℤΔ/G be the canonical map and write each vertex of ℤΔ as τ^{r}x with r ∈ ℤ and x ∈ Δ_{0}. We claim that T = Thick(T) for T = ∐_{x∈Δ_{0}} π(x). Thus we fix an indecomposable object X = π(y) in T and need to show that X belongs to Thick(T). Define for y = τ^{r}x

d(y) = \begin{cases}2r, & if x is maximal, \\ 2r − 1, & if x is minimal, \end{cases}

and use induction on d = d(y) as follows. The cases d = 0 and d = −1 are clear. Suppose now d > 0 and consider the Auslander–Reiten triangle X → Y → τ^{−1}X → ΣX starting at X. Then each indecomposable direct summand of Y is of the form π(y′) with an arrow y → y′ in ℤΔ, by Lemma A.6. We have d(y′) = d − 1 and d(τ^{−1}y) = d − 2. Thus Y and τ^{−1}X belong to Thick(T), and it follows that X is in Thick(T). The case d < 0 is similar.

The finiteness of T(T) follows from Proposition 4.5. □

5. Simply connected triangulated categories

Let T be a triangulated category that is essentially small and locally finite. We say that T is simply connected if the Auslander–Reiten quiver of T is connected and has no oriented cycle. Here, an oriented cycle is a path of length > 0 starting and ending at the same vertex. Note that there exists an oriented cycle if and only if there is a chain of non-invertible non-zero morphisms X_{0} → X_{1} → ⋯ → X_{n} = X_{0} between indecomposable objects; see Proposition 3.7.

Rings of finite representation type. For any ring A, we denote by mod A the category of finitely presented A-modules. Recall that A has finite representation type if A is artinian and there are only finitely many isomorphism classes of finitely presented indecomposable A-modules.

Theorem 5.1. For a triangulated category T the following are equivalent.

(1) The triangulated category T is essentially small, algebraic, locally finite, and simply connected.

(2) There exists a connected hereditary artinian ring A of finite representation type such that T is equivalent to the bounded derived category D^{b}(mod A).

In this case the Auslander–Reiten quiver of T is of the form ℤΔ for some tree Δ of Dynkin type.

Let us explain how the ring A is obtained from T. A species S = (K_{i},_{i}E_{j})_{i,j∈I} consists of a family of division rings K_{i} and a family of K_{i} − K_{j}-bimodules _{i}E_{j}. There is an associated tensor algebra T (S) = ⨁_{n≥0} M^{n}, where

M^{0} = ∏_{i∈I} K_{i}, M^{1} = ⨁_{i,j∈I} _{i}E_{j}, M^{n} = M^{1} ⊗_{M^{0}} … ⊗_{M^{0}} M^{1} for n > 1,

and multiplication is induced by the tensor product.

Let T be a triangulated category and suppose that T is locally finite and simply connected. Then the Auslander–Reiten quiver is of the form ℤΔ for some finite tree by Lemma 4.7. In fact, we may assume that Δ has no path of length > 1, since ℤΔ does not depend on the orientation of Δ. Choose indecomposable objects T_{i} for each vertex i ∈ Δ_{0}. Define K_{i} = End_{T}(T_{i}) and _{i}E_{j} = Hom_{T}(T_{j}, T_{i}) for i, j ∈ Δ_{0}. Observe that each K_{i} is a division ring by Proposition 3.7, and that each bimodule iE_{j} is of finite length on either side by Proposition 2.3. Then the tensor algbra corresponding to the species (K_{i},_{i}E_{j})_{i,j∈Δ_{0}} is isomorphic to the endomorphism ring of T = ∐_{i∈Δ_{0}} T_{i}. The proof is straightforward, using the fact that there are no paths of length > 1 in Δ. We denote this ring by A. Observe that A is hereditary and artinian, since K = ∏_{i∈Δ_{0}} K_{i} is semisimple and ⨁_{i,j∈Δ_{0}} _{i}E_{j} is finitely generated over K on either side.

Lemma 5.2. The object T is a tilting object, that is, T is generated by T and Hom_{T}(T, Σ^{n}T) = 0 for all n ≠ 0.

Proof. We use Proposition 3.7 which gives a path in the Auslander–Reiten quiver for each non-zero morphism between indecomposable objects. In particular, there is for each indecomposable object X a path X → Y → τ^{−1}X → ⋯ → ΣX, provided there is a non-zero morphism starting in X which is not a section.

Suppose that Hom_{T}(T_{i}, Σ^{n}T_{j}) ≠ 0 and consider the following two cases:

n > 0. From Remark 3.3 one has Hom_{T}(Σ^{n−1}T_{j}, τT_{i}) ≠ 0. Thus there is a path

T_{j} → ⋯ → Σ^{n−1}T_{j} → ⋯ → τT_{i} → T̄_{i} → T_{i}

in ℤΔ which contradicts the choice of T_{i}, T_{j} ∈ Δ_{0}. n < 0. There is a path

T_{i} → ⋯ → Σ^{n}T_{j} → ⋯ → τT_{j} → T̄_{j} → T_{j}

in ℤΔ which contradicts the choice of T_{i}, T_{j} ∈ Δ_{0}. It follows that n = 0. The fact that T = Thick(T) follows from the proof of Theorem 4.6. □

Proof of Theorem 5.1. (1) ⇒ (2): The assumptions on T yield a tilting object T = ∐_{i∈Δ_{0}} T_{i}, and its endomorphism ring A = End_{T}(T) is hereditary artinian. It follows that there are equivalences

T ⟵^{∼} K^{b}(add T) ⟶^{∼} D^{b}(mod A).

For the first equivalence, see [32, 2.1], while the second is clear from the fact that Hom_{T}(T, −) identifies add T with the category of finitely generated projective Amodules. Locally finiteness of T implies that A is of finite representation type.

(2) ⇒ (1): The abelian category mod A is hereditary, that is, Ext_{A}^{p}(−, −) vanishes for p > 1. It follows that each indecomposable complex is concentrated in a single degree. Finite representation type of A implies that T is locally finite, by Proposition 2.3. The Auslander–Reiten triangles in D^{b}(mod A) are obtained from almost split seqences in mod A. Thus the Auslander–Reiten quiver is of the form ℤΔ for some tree Δ; see [24, §4] for details. In particular, the quiver has no oriented cycles.

For the shape of Δ, see [18, §4], where hereditary rings of finite representation type are discussed. □

6. Thick subcategories and non-crossing partitions

Let A be a hereditary artin algebra. Thus A is an artinian ring that is finitely generated over its centre and Ext_{A}^{p}(−, −) = 0 for all p > 1. Note that the centre of a hereditary artin algebra is semisimple. The algebra A is said to be connected if the centre of A is a field. From now on assume that A is connected and we denote the centre by k.

In this section, we associate to A a poset of non-crossing partitions and establish a correspondence between this poset and the lattice of thick subcatgories of D^{b}(mod A). This correspondence was first observed for path algebras of finite and affine type by Ingalls and Thomas [29]. For an introduction to non-crossing partitions, see [2].

The Weyl group. Fix a representative set S_{1}, …, S_{n} of simple A-modules. We associate to A a generalised Cartan matrix C(A) = (C_{ij})_{1≤i,j≤n} as follows. Given two simple modules S_{i}, S_{j}, we have Ext_{A}^{1}(S_{i}, S_{j}) = 0 or Ext_{A}^{1}(S_{j}, S_{i}) = 0. Assume i ≠ j and Ext_{A}^{1}(S_{j}, S_{i}) = 0. Then define

C_{ij} = −ℓ_{End_{A}(S_{i})}(Ext_{A}^{1}(S_{i}, S_{j})) and C_{ji} = −ℓ_{End_{A}(S_{j})}(Ext_{A}^{1}(S_{i}, S_{j})).

In addition, define C_{ii} = 2 and d_{i} = ℓ_{k}(End_{A}(S_{i})) for each i. Then one has d_{i}C_{ij} = d_{j}C_{ji}. Thus C(A) is a symmetrisable generalised Cartan matrix in the sense of [31].

Next we consider the Weyl group corresponding to C(A). Let ℝ^{n} denote the ndimensional real space with standard basis ε_{1}, …, ε_{n}. Define a symmetric bilinear form by (ε_{i}, ε_{j}) = d_{i}C_{ij} and for each α ∈ ℝ^{n} with (α, α) ≠ 0 the reflection

s_{α}: ℝ^{n} ⟶ ℝ^{n}, ξ ↦ ξ − 2\frac{(ξ, α)}{(α, α)}α.

We write s_{i} for the simple reflection s_{ε_{i}} and observe that each s_{i} maps ℤ^{n} into itself. The Weyl group is the group W generated by the simple reflections s_{1}, …, s_{n}. The real roots are by definition the elements of ℤ^{n} of the form w(ε_{i}) for some w ∈ W and some i ∈ {1, …, n}. Note that for any real root α the corresponding reflection s_{α} belongs to W since s_{w(α)} = ws_{α}w^{−1}.

We define the absolute order on W with respect to the absolute length ℓ as follows. Consider the set of reflections

W_{1} = {ws_{i}w^{−1} | w ∈ W, 1 ≤ i ≤ n}

and for each w ∈ W let ℓ(w) denote the minimal r ≥ 0 such that w can be written as product w = x_{1} … x_{r} of reflections x_{j} ∈ W_{1}. Given u, v ∈ W define

u ≤ v ⟺ ℓ(u) + ℓ(u^{−1}v) = ℓ(v).

Note that the length function ℓ and the absolute order are invariant under conjugation with a fixed element of W.

A Coxeter element in W is any element of W that is conjugate to one of the form s_{σ(1)}s_{σ(2)} … s_{σ(n)} for some permutation σ. Note that ℓ(c) = n for each Coxeter element c by [19, Theorem 1.1]. This has the following immediate consequence.

Lemma 6.1. Let c be a Coxeter element and x_{1}, …, x_{n} a sequence of reflections in W_{1} such that c = x_{1} ⋯ x_{n}. If 1 ≤ r ≤ s ≤ n, then

ℓ(x_{1} … x_{r}) = r and x_{1} … x_{r} ≤ x_{1} … x_{s}.  □

Relative to a Coxeter element c one defines the poset of non-crossing partitions

NC(W, c) = {w ∈ W | id ≤ w ≤ c}.

Given two Coxeter elements c, c′ in W, we have NC(W, c) ≅ NC(W, c′) provided that c and c′ are conjugate.

The Grothendieck group K_{0}(mod A) is isomorphic to ℤ^{n} via the map sending each simple A-module S_{i} to the standard base vector ε_{i}. The image of an Amodule X under this map is called dimension vector and denoted by dim X; we write s_{X} = s_{dim X} for the corresponding reflection.

Exceptional modules and sequences. An A-module X is called exceptional if X is indecomposable and Ext_{A}^{1}(X, X) = 0. Note that dim X is a real root if X is exceptional [45, Corollary 2]. A sequence (X_{1}, …, X_{r}) of A-modules is called exceptional if each X_{i} is exceptional and

Hom_{A}(X_{j}, X_{i}) = 0 = Ext_{A}^{1}(X_{j}, X_{i}) for all i < j.

Such a sequence is complete if r = n.

Lemma 6.2. Sending an A-module X to the reflection s_{X} = s_{dim X} gives an injective map from the set of isomorphism classes of exceptional A-modules into W.

Proof. An exceptional A-module is uniquely determined by its dimension vector; see [35, Lemma 8.2]. On the other hand, given a reflection s ∈ W_{1}, we have s = s_{α} where α ∈ ℤ^{n} is the unique vector with non-negative entries satisfying s(α) = −α. Thus s_{dim X} = s_{dim Y} implies X ≅ Y. □

Theorem 6.3 (Crawley-Boevey, Ringel, Igusa–Schiffler). Let A be a connected hereditary artin algebra with simple modules S_{1}, …, S_{n} satisfying Ext_{A}^{1}(S_{j}, S_{i}) = 0 for all i < j. Denote by W the associated Weyl group and fix the Coxeter element c = s_{1} ⋯ s_{n}. Then the braid group B_{n} on n strings acts transitively on — the isomorphism classes of complete exceptional sequences (X_{1}, …, X_{n}) in

mod A, and

— the sequences (x_{1}, …, x_{n}) of reflections in W_{1} such that c = x_{1} ⋯ x_{n}. Moreover, σ(X_{1}, …, X_{n}) = (Y_{1}, …, Y_{n}) implies σ(s_{X_{1}}, …, s_{X_{n}}) = (s_{Y_{1}}, …, s_{Y_{n}}) for all σ ∈ B_{n}.

Proof. For the action of the braid group on complete exceptional sequences, see [17, 45]. For the action on factorisations of the Coxeter element, see [28, Theorem 1.4]. The compatibility of both actions follows from the computation of the dimension vectors of the modules in an exceptional sequence under the braid group action; see the proof of [28, Corollary 2.4] and also the Corollary in [17]. □

Corollary 6.4. Let (x_{1}, …, x_{n}) be a sequence of reflections in W_{1} such that c = x_{1} ⋯ x_{n}. Then there exists up to isomorphism a unique complete exceptional sequence (X_{1}, …, X_{n}) such that x_{i} = s_{X_{i}} for all i.

Proof. Theorem 6.3 gives σ ∈ B_{n} such that (x_{1}, …, x_{n}) = σ(s_{1}, …, s_{n}). Let (X_{1}, …, X_{n}) = σ(S_{1}, …, S_{n}). Then x_{i} = s_{X_{i}} for all i. Uniqueness follows from Lemma 6.2. □

Thick subcategories. We consider the bounded derived category D^{b}(mod A) and identify A-modules with complexes concentrated in degree zero. Recall the following correspondence.

Proposition 6.5 (Brüning). Let A be a hereditary abelian category. The canonical inclusion A → D^{b}(A) induces a bijection between

– the set of exact abelian and extension closed subcategories of A, and

– the set of thick subcategories of D^{b}(A). The bijection sends C ⊆ A to {X ∈ D^{b}(A) | H^{i}X ∈ C for all i ∈ ℤ}. Its inverse sends U ⊆ D^{b}(A) to

H^{0}U = {Y ∈ A | Y ≅ H^{0}X for some X ∈ U}.

Proof. See [14, Theorem 5.1]. □

The Loewy length of an object X in some abelian category is the smallest p ≥ 0 such that there exists a chain 0 = X_{0} ⊆ X_{1} ⊆ … ⊆ X_{p} = X so that X_{i+1}/X_{i} is semisimple for all i. The height of an abelian category is the supremum of the Loewy lengths of its objects.

Next we characterise the thick subcategories of D^{b}(mod A) such that the inclusion admits an adjoint. The connection with exceptional sequences is due to Bondal [12].

Proposition 6.6. Let A be a hereditary artin algebra. For a thick subcategory U of D^{b}(mod A) are equivalent:

(1) The inclusion U → D^{b}(mod A) admits a left adjoint.

(2) The inclusion H^{0}U → mod A admits a left adjoint.

(3) The abelian category H^{0}U is of finite height and has only finitely many isomorphism classes of simple objects.

(4) There exists an exceptional sequence (X_{1}, …, X_{r}) in mod A such that U = Thick(X_{1}, …, X_{r}).

(5) There exists a complete exceptional sequence (X_{1}, …, X_{n}) in mod A such that

U = Thick(X_{1}, …, X_{r}) and ^{⊥}U = Thick(X_{r+1}, …, X_{n})

Remark 6.7. The k-duality (mod A)^{op} ⟶^{∼} mod(A^{op}) induces a duality

D^{b}(mod A)^{op} ⟶^{∼} D^{b}(mod A^{op})

which preserves the property (3). Thus the existence of left adjoints in (1) and (2) is equivalent to the existence of right adjoints.

Remark 6.8. The equivalent conditions in Proposition 6.6 are automatically satisfied if the algebra A is of finite representation type; see Theorem 2.5.

Proof of Proposition 6.6. (1) ⇔ (2): See [40, §2].

(2) ⇒ (3): A left adjoint F: mod A → H^{0}U sends a projective generator to a projective generator. Thus one takes B = End_{A}(F A) and gets an equivalence

Hom_{A}(F A, −): H^{0}U ⟶^{∼} mod B.

Clearly, mod B has finite height and only finitely many simple objects.

(3) ⇒ (4): It follows from [22, 8.2] that the category H^{0}U is equivalent to mod B for some finite dimensional k-algebra B. The algebra B is hereditary since A is hereditary. Thus the simple B-modules form a complete exceptional sequence (X_{1}, …, X_{r}) in mod B. This gives an exceptional sequence in mod A satisfying U = Thick(X_{1}, …, X_{r}).

(4) ⇒ (1): See [12, Theorem 3.2].

(1)–(4) ⇒ (5): The duality provides a right adjoint for the inclusion of U. Thus the inclusion of^{⊥}U admits a left adjoint by Lemma 2.4. It follows that there is an exceptional sequence (X_{r+1}, …, X_{s}) in mod A with^{⊥}U = Thick(X_{r+1}, …, X_{s}). Then (X_{1}, …, X_{s}) is an exceptional sequence satisfying Thick(X_{1}, …, X_{s}) = D^{b}(mod A) which is therefore complete.

(5) ⇒ (4): Clear. □

Example 6.9. For a tame hereditary algebra A, the regular A-modules form an exact abelian and extension closed subcategory of mod A that is of infinite height.

A classification. The following result is a consequence of Theorem 6.3 and provides a combinatorial classification of the thick subcategories of D^{b}(mod A) satisfying the equivalent conditions in Proposition 6.6. Using the correspondence in Proposition 6.5, this translates into a classification of certain abelian subcategories of mod A.

For the path algebra of a quiver over an algebraically closed field, this result is due to Igusa, Schiffler, and Thomas [28], Previous work of Ingalls and Thomas [29] establishes the result in finite and affine type.

Theorem 6.10. Let A be a connected hereditary artin algebra with simple modules S_{1}, …, S_{n} satisfying Ext_{A}^{1}(S_{j}, S_{i}) = 0 for all i < j. Denote by W the associated Weyl group and fix the Coxeter element c = s_{1} ⋯ s_{n}. Then there exists an order preserving bijection between

— the set of thick subcategories of D^{b}(mod A) such that the inclusion admits a left adjoint, and

— the set of non-crossing partitions NC(W, c). The map sends a thick subcategory which is generated by an exceptional sequence (X_{1}, …, X_{r}) to s_{X_{1}} ⋯ s_{X_{r}}.

Let us formulate an immediate consequence.

Corollary 6.11. Let (X_{1}, …, X_{r}) and (Y_{1}, …, Y_{s}) be exceptional sequences in mod A. Then

Thick(X_{1}, …, X_{r}) = Thick(Y_{1}, …, Y_{s}) ⟺ s_{X_{1}} ⋯ s_{X_{r}} = s_{Y_{1}} ⋯ s_{Y_{s}}. □

Proof of Theorem 6.10. Fix a thick subcategory U ⊆ D^{b}(mod A) such that the inclusion admits a left adjoint. There exists a complete exceptional sequence (X_{1}, …, X_{n}) in mod A such that U = Thick(X_{1}, …, X_{r}) for some r ≤ n, by Proposition 6.6. We assign to U the element cox(U) = s_{X_{1}} ⋯ s_{X_{r}} in W. Observe that cox(U) ≤ c by Lemma 6.1, since s_{X_{1}} ⋯ s_{X_{n}} = c by Theorem 6.3. Thus cox gives a map into NC(W, c).

The map cox is well-defined: Choose a second exceptional sequence (Y_{1}, …, Y_{s}) in mod A such that U = Thick(Y_{1}, …, Y_{s}). Then (Y_{1}, …, Y_{s}, X_{r+1}, …, X_{n}) is a complete exceptional sequence, and it follows from Theorem 6.3 that

s_{Y_{1}} ⋯ s_{Y_{s}}s_{X_{r+1}} ⋯ s_{X_{n}} = c = s_{X_{1}} ⋯ s_{X_{r}}s_{X_{r+1}} ⋯ s_{X_{n}}.

Thus s_{Y_{1}} ⋯ s_{Y_{s}} = s_{X_{1}} ⋯ s_{X_{r}}.

The map cox is injective: Let U and V be thick subcategories such that cox(U) = cox(V). Thus there are two complete exceptional sequences (X_{1}, …, X_{n}) and (Y_{1}, …, Y_{n}) such that U = Thick(X_{1}, …, X_{r}) and V = Thick(Y_{1}, …, Y_{s}) for some pair of integers r, s ≤ n. Moreover, s_{X_{1}} ⋯ s_{X_{r}} = s_{Y_{1}} ⋯ s_{Y_{s}}. It follows that

r = ℓ(s_{X_{1}} ⋯ s_{X_{r}}) = ℓ(s_{Y_{1}} ⋯ s_{Y_{s}}) = s

and therefore c = s_{Y_{1}} ⋯ s_{Y_{s}}s_{X_{r+1}} ⋯ s_{X_{n}}. From Corollary 6.4 and Lemma 6.2 one gets that (Y_{1}, …, Y_{s}, X_{r+1}, …, X_{n}) is a complete exceptional sequence. Thus

U = Thick(X_{1}, …, X_{r}) = Thick(X_{r+1}, …, X_{n})^{⊥} = Thick(Y_{1}, …, Y_{s}) = V.

The map cox is surjective: Let w ∈ NC(W, c). Thus we can write

c = x_{1} ⋯ x_{r}x_{r+1} ⋯ x_{n}

as products of reflections x_{i} ∈ W_{1} such that w = x_{1} ⋯ x_{r}. From Corollary 6.4 one gets a complete exceptional sequence (X_{1}, …, X_{n}) with x_{i} = s_{X_{i}} for all i. Set U = Thick(X_{1}, …, X_{r}). Then cox(U) = w.

The map cox is order preserving: Let V ⊆ U be thick subcategories. From Proposition 6.6 one gets a complete exceptional sequence (X_{1}, …, X_{n}) such that

V = Thick(X_{1}, …, X_{s}) and U = Thick(X_{1}, …, X_{r})

for some s ≤ r ≤ n. It follows from Lemma 6.1 that

cox(V) = s_{X_{1}} ⋯ s_{X_{s}} ≤ s_{X_{1}} ⋯ s_{X_{r}} = cox(U).  □

Remark 6.12. Let U ⊆ D^{b}(mod A) be a thick subcategory such that the inclusion admits a left adjoint. Then cox(U) cox(^{⊥}U) = c.

Example: The Kronecker algebra. Let k be a field and consider the Kronecker algebra, that is, the path algebra of the quiver \xymatrix{∘ \ar[r] \ar[r] & ∘}. This is a tame hereditary artin algebra; we denote it by K and compute the lattice of thick subcategories of D^{b}(mod K). For a description of mod K, we refer to [6, VIII.7].

Each finite dimensional indecomposable K-module is either exceptional or regular. The dimension vectors of the exceptional K-modules are (p, q) ∈ ℤ^{2} with p, q ≥ 0 and |p − q| = 1. Thus the non-crossing partitions with respect to the Coxeter element c = s_{1}s_{2} form the lattice

NC(W, c) = {s_{(p,q)} | p, q ≥ 0 and |p − q| = 1} ∪ {id, c}

with the following Hasse diagram:

[figure: c; ⋯; •; •; •; ⋯; id]

The regular K-modules form an extension closed exact abelian subcategory of mod K that is uniserial. The simple objects of this abelian category are parameterised by the closed points of the projective line over k, which we identify with non-maximal and non-zero homogeneous prime ideals 𝔭 ⊆ k[x, y]. We denote the set of closed points by ℙ^{1}(k) and write 2^{ℙ^{1}(k)} for its power set. Adding an extra greatest element (the set of all non-maximal homogeneous prime ideals) to 2^{ℙ^{1}(k)} yields a new lattice which we denote by ˆ2^{ℙ^{1}(k)}. The simple object corresponding to 𝔭 is denoted by S_{𝔭}. We obtain an injective map

ˆ2^{ℙ^{1}(k)} ⟶ T(D^{b}(mod K))

by sending U ⊆ ℙ^{1}(k) to Thick({S_{𝔭} | 𝔭 ∈ U}) and ℙ^{1}(k) ∪ {0} to D^{b}(mod K).

Let L′, L′′ be a pair of lattices with smallest elements 0′, 0′′ and greatest elements 1′, 1′′. Denote by L′ ⨿ L′′ the new lattice which is obtained from the disjoint union L′ ∪ L′′ (viewed as sum of posets) by identifying 0′ = 0′′ and 1′ = 1′′.

Proposition 6.13. The lattice of thick subcategories of D^{b}(mod K) is isomorphic to the lattice NC(W, c) ⨿ ˆ2^{ℙ^{1}(k)}.

Proof. We write T(K) = T(D^{b}(mod K)) and have injective maps

NC(W, c) ⟶ T(K) and ˆ2^{ℙ^{1}(k)} ⟶ T(K)

which induce an order preserving map

NC(W, c) ⨿ ˆ2^{ℙ^{1}(k)} ⟶ T(K).

In order to prove bijectivity, fix a pair of indecomposable K-modules X, Y such that X is exceptional and Y is regular. We have Thick(X)∩Thick(Y) = 0 and this gives injectivity; surjectivity follows from the fact that Thick(X, Y) = D^{b}(mod K). □

The category of coherent sheaves on the projective line ℙ_{k}^{1} admits a tilting object T = O(0) ⊕ O(1) such that End(T) ≅ K; see [8]. Thus RHom(T, −) induces a triangle equivalence

D^{b}(coh ℙ_{k}^{1}) ⟶^{∼} D^{b}(mod K)

and this yields a description of the lattice of thick subcategories of D^{b}(coh ℙ_{k}^{1}). Note that the category of coherent sheaves carries a tensor product. The thick tensor subcategories have been classified by Thomason [46, Theorem 3.5]; they are precisely the ones parameterised by subsets of ℙ^{1}(k).

Appendix A. Auslander–Reiten theory

In this appendix we collect some basic facts from Auslander–Reiten theory, as initiated by Auslander and Reiten in [5]. Krull–Remak–Schmidt categories form the appropriate setting for this theory, while exact or triangulated structures are irrelevant for most parts; see also [7, 41]. This material is well-known, at least for categories that are linear over a field with finite dimensional morphism spaces. We provide full proofs for most statements, including references whenever they are available.

Let C be an essentially small additive category. Then C is called Krull–Remak– Schmidt category if every object in C is a finite coproduct of indecomposable objects with local endomorphism rings.

It is convenient to view C as a ring with several objects. Thus we use the category Mod C of C-modules, which are by definition the additive functors C^{op} → Ab into the category of abelian groups.

There is the following useful characterisation in terms of projective covers. Recall that a morphism ϕ: P → M is a projective cover, if P is a projective object and ϕ is an essential epimorphism, that is, a morphism α: P′ → P is an epimorphism if and only if ϕα is an epimorphism.

Lemma A.1. For an essentially small additive category C with split idempotents the following are equivalent:

(1) The category C is a Krull–Remak–Schmidt category.

(2) Every finitely generated C-module admits a projective cover.

Proof. First observe that the Yoneda functor C → Mod C taking an object X to Hom_{C}(−, X) identifies C with the category of finitely generated projective Cmodules. Next observe that an epimorphism ϕ: P → M in Mod C is a projective cover if and only if every endomorphism α: P → P satisfying ϕα = ϕ is an isomorphism. If M is simple this is equivalent to End_{C}(P) being a local ring.

(1) ⇒ (2): Let M be a simple C-module and choose an indecomposable object X with M(X) ≠ 0. Then there exists a non-zero morphism Hom_{C}(−, X) → M which is a projective cover since End_{C}(X) is local. Thus every finite sum of simple C-module admits a projective cover.

Now let M be a finitely generated C-module. Denote by rad M its radical, that is, the intersection of all maximal submodules of M. Choose an epimorphism ϕ: P → M with P finitely generated projective. Let P = ⨁_{i} P_{i} be a decomposition into indecomposable modules. Then

P/ rad P = ⨁_{i} P_{i}/ rad P_{i}

is a finite sum of simple C-modules since each P_{i} has a local endomorphism ring. The epimorphism ϕ induces an epimorphism P/ rad P → M/ rad M and therefore M/ rad M decomposes into finitely many simple modules. There exists a projective cover Q → M/ rad M and this factors through the canonical morphism π: M → M/ rad M via a morphism ϕ: Q → M. The morphism π is essential since M is finitely generated. It follows that ϕ is an epimorphism. The morphism ϕ is essential since πϕ is essential. Thus ϕ is a projective cover.

(2) ⇒ (1): Let X be an object in C. Then X admits a decomposition into finitely many indecomposable objects. Thus we need to show that End_{C}(X) is local if X is indecomposable. Choose a maximal subobject U ⊆ P = Hom_{C}(−, X). Then S = P/U is simple and admits a projective cover P′ → S. It is easily checked that P ≅ P′ since P is indecomposable. Thus End_{C}(X) ≅ End_{C}(P) is a local ring. □

Remark A.2. Fix a ring A and let C be the category of finitely generated projective A-modules. Then the functor Mod C → Mod A taking X to X(A) is an equivalence. The category C is Krull–Remak–Schmidt if and only if the ring A is semi-perfect.

From now on suppose that C is a Krull–Remak–Schmidt category.

The radical. The radical of C is by definition the collection of subgroups

Rad_{C}(X, Y) ⊆ Hom_{C}(X, Y)

for each pair X, Y of objects in C, where

Rad_{C}(X, Y) = {ϕ ∈ Hom_{C}(X, Y) | id_{X} −ϕ′ϕ is invertible for all ϕ′: Y → X}

= {ϕ ∈ Hom_{C}(X, Y) | id_{Y} −ϕϕ′ is invertible for all ϕ′: Y → X}.

The radical is a two-sided ideal of C, that is, a subfunctor of Hom_{C}(−, −): C^{op}×C → Ab. It is actually the unique two-sided ideal 𝔍 of C such that 𝔍(X, X) equals the Jacobson radical of End_{C}(X) for each object X; see [34].

Given two decompositions X = ∐_{i} X_{i} and Y = ∐_{j} Y_{j} in C, we have

Rad_{C}(X, Y) = ⨁_{i,j} Rad_{C}(X_{i}, Y_{j}).

A morphism between indecomposable objects belongs to the radical if and only if it is not invertible, since indecomposable objects have local endomorphism rings.

Simple functors. Given an object X in C, the functor Rad_{C}(−, X) equals the intersection of all maximal subfunctors of Hom_{C}(−, X): C^{op} → Ab. This observation has the following consequence, where

S_{X} = Hom_{C}(−, X)/ Rad_{C}(−, X).

Lemma A.3 ([3, Proposition 2.3]). The map sending an object X of C to the functor S_{X} induces (up to isomorphism) a bijection between the indecomposable objects of C and the simple objects of Mod C.

Proof. If an object X has a local endomorphism ring, then Rad_{C}(−, X) is the unique maximal subobject of Hom_{C}(−, X). Thus S_{X} is simple in that case, and the inverse map sends a simple object S in Mod C to the unique indecomposable object X in C such that S(X) ≠ 0. □

Irreducible morphism. The definition of the radical is extended recursively as follows. For each n > 1 and each pair of objects X, Y let Rad_{C}^{n}(X, Y) be the set of morphisms ϕ ∈ Hom_{C}(X, Y) that admit a factorisation ϕ = ϕ′′ϕ′ with ϕ′ ∈ Rad_{C}(X, Z) and ϕ′′ ∈ Rad_{C}^{n−1}(Z, Y) for some object Z. Then we set

Irr_{C}(X, Y) = Rad_{C}(X, Y)/ Rad_{C}^{2}(X, Y).

This is a bimodule over the division rings Δ(X) and Δ(Y), where

Δ(X) = End_{C}(X)/ Rad_{C}(X, X).

Note that a morphism X → Y between indecomposable objects belongs to Rad_{C}(X, Y) ∖ Rad_{C}^{2}(X, Y) if and only it is irreducible. A morphism ϕ is called irreducible if ϕ is neither a section nor a retraction, and if ϕ = ϕ′′ϕ′ is a factorisation then ϕ′ is a section or ϕ′′ is a retraction.

Lemma A.4. For each pair of indecomposable objects X, Y in C, we have

Ext_{C}^{1}(S_{Y} , S_{X}) ≅ Hom_{Δ(X)}(Irr_{C}(X, Y), Δ(X))

as bimodules over Δ(X) and Δ(Y).

Proof. Applying Hom_{C}(−, S_{X}) to the exact sequence

0 ⟶ Rad_{C}(−, Y) ⟶ Hom_{C}(−, Y) ⟶ S_{Y} ⟶ 0

gives

Ext_{C}^{1}(S_{Y} , S_{X}) ≅ Hom_{C}(Rad_{C}(−, Y), S_{X}).

Then applying Hom_{C}(−, S_{X}) to the exact sequence

0 ⟶ Rad_{C}^{2}(−, Y) ⟶ Rad_{C}(−, Y) ⟶ Irr_{C}(−, Y) ⟶ 0

gives

Ext_{C}^{1}(S_{Y} , S_{X}) ≅ Hom_{C}(Irr_{C}(−, Y), S_{X}) ≅ Hom_{Δ(X)}(Irr_{C}(X, Y), Δ(X))

since S_{X}(X) = Δ(X). □

Almost split morphisms. A morphism ϕ: X → Y is called right almost split if ϕ is not a retraction and if every morphism X′ → Y that is not a retraction factors through ϕ. The morphism ϕ is right minimal if every endomorphism α: X → X with ϕα = ϕ is invertible. Note that Y is indecomposable if ϕ is right almost split. Left almost split morphisms and left minimal morphisms are defined dually. The term minimal right almost split means right minimal and right almost split.

Lemma A.5 ([4, Chap. II, Proposition 2.7]). A morphism X → Y in C is minimal right almost split if and only if it induces in Mod C a minimal projective presentation

Hom_{C}(−, X) ⟶ Hom_{C}(−, Y) ⟶ S ⟶ 0

of a simple object.

Proof. A non-zero morphism Hom_{C}(−, Y) → S to a simple object is a projective cover if and only if End_{C}(Y) is local. In that case the exactness of the sequence means that the image of Hom_{C}(−, X) → Hom_{C}(−, Y) equals Rad_{C}(−, Y); it is therefore equivalent to the fact that X → Y is right almost split. The canonical morphism from Hom_{C}(−, X) to the image of Hom_{C}(−, X) → Hom_{C}(−, Y) is a projective cover if and only if X → Y is right minimal. □

Almost split morphisms and irreducible morphisms are related as follows.

Lemma A.6 ([7, Corollary 3.4]). Let X → Y be a minimal right almost split morphism in C and let X = X_{1}^{n_{1}} ⨿…⨿X_{r}^{n_{r}} be a decomposition into indecomposable objects such that the X_{i} are pairwise non-isomorphic. Given an indecomposable object X′, one has Irr_{C}(X′, Y) ≠ 0 if and only if X′ ≅ X_{i} for some i. In that case n_{i} equals the length of Irr_{C}(X′, Y) over Δ(X′).

Proof. The morphism X → Y induces a minimal projective presentation

Hom_{C}(−, X) ⟶ Hom_{C}(−, Y) ⟶ S_{Y} ⟶ 0

by Lemma A.5, and therefore a projective cover

π: Hom_{C}(−, X) ⟶ Rad_{C}(−, Y).

This morphism induces an isomorphism

Hom_{C}(−, X)/ Rad_{C}(−, X) ⟶^{∼} Rad_{C}(−, Y)/ Rad_{C}^{2}(−, Y)

since Ker π ⊆ Rad_{C}(−, X). On the other hand, the decomposition of X implies

Hom_{C}(−, X)/ Rad_{C}(−, X) ≅ S_{X_{1}}^{n_{1}} ⨿ … ⨿ S_{X_{r}}^{n_{r}}.

It remains to observe that S_{X_{i}}(X′) ≠ 0 if and only if X′ ≅ X_{i}. □

Almost split sequences. The following definition of an almost split sequence is taken from Liu [41]; it covers the original definition of Auslander and Reiten for abelian categories [5], but also Happel’s definition of an Auslander–Reiten triangle in a triangulated category [24].

A sequence of morphisms X ⟶^{α} Y ⟶^{β} Z in C is called almost split if

(1) α is minimal left almost split and a weak kernel of β,

(2) β is minimal right almost split and a weak cokernel of α, and

(3) Y ≠ 0.

The end terms X and Z determine each other up to isomophism, and we write τX = Z and τ^{−1}Z = X. One calls τZ the Auslander–Reiten translate of Z.

Lemma A.7. A sequence of morphisms X → Y → Z in C is almost split if and only if it induces two minimal projective presentations

Hom_{C}(−, X) → Hom_{C}(−, Y) → Hom_{C}(−, Z) → S_{Z} → 0
Hom_{C}(Z, −) → Hom_{C}(Y, −) → Hom_{C}(X, −) → S^{X} → 0

where we use the notation

S^{X} = Hom_{C}(X, −)/ Rad_{C}(X, −).

Proof. Apply Lemma A.5. □

The Auslander–Reiten translate is functorial in the following sense.

Lemma A.8. Let X, Y be indecomposable objects in C and suppose their Auslander– Reiten translates are defined. Then Δ(X) ≅ Δ(τX) and Δ(Y) ≅ Δ(τY) as division rings. Using these isomorphisms, we have

Hom_{Δ(X)}(Irr_{C}(X, Y), Δ(X)) ≅ Hom_{Δ(τY)}(Irr_{C}(τX, τY), Δ(τY))

as bimodules over Δ(X) and Δ(Y).

Proof. We use the stable category of finitely presented C-modules which we denote by \underline{mod} C. The objects are the C-modules M that admit a presentation

(∗)  Hom_{C}(−, X) ⟶ Hom_{C}(−, Y) ⟶ M ⟶ 0

and the morphisms are all C-linear morphisms modulo the subgroup of morphisms that factor through a projective C-module. There are two functors

Ω: \underline{mod} C ⟶ \underline{mod} C and Tr: \underline{mod} C ⟶ \underline{mod} C^{op}

taking a module to its syzygy and its transpose, respectively. Both functors are defined for a module M with presentation (∗) via exact sequences as follows:

0 ⟶ ΩM ⟶ Hom_{C}(−, Y) ⟶ M ⟶ 0
Hom_{C}(Y, −) ⟶ Hom_{C}(X, −) ⟶ Tr M ⟶ 0

Note that the transpose yields an equivalence (\underline{mod} C)^{op} ⟶^{∼} \underline{mod} C^{op}.

Using the minimal projective presentations of S_{X} and S^{τX} from Lemma A.7, one gets

Δ(τX) ≅ End_{C}(S^{τX}) ≅ End_{C}(S_{X}) ≅ Δ(X).

The same minimal presentations yield isomorphisms

ΩS_{X} ≅ Tr S^{τX}  and  ΩS^{τX} ≅ Tr S_{X}.

In fact, the minimality of these presentations implies that ΩS_{X} = Rad_{C}(−, X) has no non-zero projective direct summand. From these isomorphisms and Lemma A.4, we obtain

Hom_{Δ(X)}(Irr_{C}(X, Y), Δ(X)) ≅ Ext_{C}^{1}(S_{Y} , S_{X})
≅ \underline{Hom}_{C}(ΩS_{Y} , S_{X})
≅ \underline{Hom}_{C}(Tr S^{τY} , S_{X})
≅ \underline{Hom}_{C^{op}}(Tr S_{X}, S^{τY})
≅ \underline{Hom}_{C^{op}}(ΩS^{τX}, S^{τY})
≅ Ext_{C^{op}}^{1}(S_{τX}, S_{τY})
≅ Hom_{Δ(τX)}(Irr_{C^{op}}(τY, τX), Δ(τX))
≅ Hom_{Δ(τX)}(Irr_{C}(τX, τY), Δ(τX)).

The second isomorphism requires an extra argument. Let ΩS_{Y} = Rad_{C}(−, Y). Then we have

Ext_{C}^{1}(S_{Y} , S_{X}) ≅ Hom_{C}(ΩS_{Y} , S_{X}) ≅ \underline{Hom}_{C}(ΩS_{Y} , S_{X}).

The first isomorphism is from the proof of Lemma A.4. The second follows from the fact that any non-zero morphism ΩS_{Y} → S_{X} factoring through a projective factors through the projective cover Hom_{C}(−, X) → S_{X}. This means that Hom_{C}(−, X) is a direct summand of ΩS_{Y}, which has been excluded before. □

The Auslander–Reiten quiver. The Auslander–Reiten quiver of C is defined as follows. The set of isomorphism classes of indecomposable objects in C form the vertices, and there is an arrow X → Y if Irr_{C}(X, Y) ≠ 0. It is often convenient to identify an indecomposable object with its isomorphism class.

The Auslander–Reiten quiver carries a valuation which assigns to each arrow X → Y the pair (δ_{X,Y} , δ′_{X,Y}), where

δ_{X,Y} = ℓ_{Δ(X)}(Irr_{C}(X, Y)) and δ′_{X,Y} = ℓ_{Δ(Y)}(Irr_{C}(X, Y)).

Here, ℓ_{A}(M) denotes the length of an A-module M.

Lemma A.9. Let X, Y be indecomposable objects in C and suppose that τY is defined. Then δ′_{τY,X} = δ_{X,Y}.

Proof. We have an almost split sequence τY → X̄ → Y and δ_{X,Y} counts the multiplicity of X in a decomposition of X̄ by Lemma A.6, which equals δ′_{τY,X} by the dual of Lemma A.6. □

Lemma A.10. Let k be a commutative ring and suppose that C is k-linear with all morphism spaces of finite length over k. Let X, Y be indecomposable objects in C and suppose that τY is defined. Then δ_{τY,X} = δ′_{X,Y}.

Proof. Using the identity δ′_{τY,X} = δ_{X,Y} from Lemma A.9 and the isomorphism Δ(Y) ≅ Δ(τY) from Lemma A.8, one computes

ℓ_{Δ(Y)}(Irr_{C}(X, Y)) = ℓ_{k}(Irr_{C}(X, Y)) · ℓ_{k}(Δ(Y))^{−1}
= ℓ_{Δ(X)}(Irr_{C}(X, Y)) · ℓ_{k}(Δ(X)) · ℓ_{k}(Δ(Y))^{−1}
= ℓ_{Δ(X)}(Irr_{C}(τY, X)) · ℓ_{k}(Δ(X)) · ℓ_{k}(Δ(Y))^{−1}
= ℓ_{k}(Irr_{C}(τY, X)) · ℓ_{k}(Δ(τY))^{−1}
= ℓ_{Δ(τY)}(Irr_{C}(τY, X)).  □

The repetition. Let Γ be a quiver without loops. Then a new quiver ℤΓ is defined as follows. The set of vertices is ℤ × Γ_{0}. For each arrow x → y in Γ and each n ∈ ℤ there is an arrow (n, x) → (n, y) and an arrow (n − 1, y) → (n, x) in ℤΓ. The quiver ℤΓ is a translation quiver [43] with translation τ defined by τ(n, x) = (n − 1, x) for each vertex (n, x).

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Henning Krause, Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.

E-mail address: hkrause@math.uni-bielefeld.de